Calculus Examples

$lim_{x \to \infty} \frac{1 - x}{2 x}$

Step 1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to \infty} \frac{1 - x}{x}$

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x$ .

$\frac{1}{2} lim_{x \to \infty} \frac{\frac{1}{x} - \frac{x}{x}}{\frac{x}{x}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

Cancel the common factor of $x$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{1}{2} lim_{x \to \infty} \frac{\frac{1}{x} - \frac{x}{x}}{\frac{x}{x}}$

Step 3.1.1.2

Rewrite the expression.

$\frac{1}{2} lim_{x \to \infty} \frac{\frac{1}{x} - 1 \cdot 1}{\frac{x}{x}}$

Step 3.1.2

Multiply $- 1$ by $1$ .

$\frac{1}{2} lim_{x \to \infty} \frac{\frac{1}{x} - 1}{\frac{x}{x}}$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Cancel the common factor.

$\frac{1}{2} lim_{x \to \infty} \frac{\frac{1}{x} - 1}{\frac{x}{x}}$

Step 3.2.2

Rewrite the expression.

$\frac{1}{2} lim_{x \to \infty} \frac{\frac{1}{x} - 1}{1}$

Step 3.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

$\frac{1}{2} \cdot \frac{lim_{x \to \infty} \frac{1}{x} - 1}{lim_{x \to \infty} 1}$

Step 3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{1}{2} \cdot \frac{lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} 1}{lim_{x \to \infty} 1}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{1}{2} \cdot \frac{0 - lim_{x \to \infty} 1}{lim_{x \to \infty} 1}$

Step 5

Evaluate the limit.

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Step 5.1

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

$\frac{1}{2} \cdot \frac{0 - 1 \cdot 1}{lim_{x \to \infty} 1}$

Step 5.2

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

$\frac{1}{2} \cdot \frac{0 - 1 \cdot 1}{1}$

Step 5.3

Simplify the answer.

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Step 5.3.1

Divide $0 - 1 \cdot 1$ by $1$ .

$\frac{1}{2} (0 - 1 \cdot 1)$

Step 5.3.2

Multiply $- 1$ by $1$ .

$\frac{1}{2} (0 - 1)$

Step 5.3.3

Subtract $1$ from $0$ .

$\frac{1}{2} \cdot - 1$

Step 5.3.4

Combine $\frac{1}{2}$ and $- 1$ .

$\frac{- 1}{2}$

Step 5.3.5

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$